Financial Engineering 6

Lecture 24

Advanced Derivatives

References: Villalobos, J.P. Morgan

Lecture Topics • Did you briefly read The J.P. Morgan Guide to Credit

Derivatives? • Credit Derivatives • Collateralized Debt Obligations • Examples • Introduction to Risk Assessment

The Need (Credit Event) From the J.P. Morgan paper, we see the following potential reasons or risks that might promote the use of a credit derivative:

• Failure to meet payment obligations when due.

• Bankruptcy.

• Repudiation – a declaration by one party in a contract stating that they do not intend to live up to their obligations.

• Material adverse restructuring of debt.

• Obligation acceleration or obligation default.

Credit Derivative • A credit derivative is a securitized derivative whose value is

derived from the credit risk on an underlying bond, loan or any other financial asset issued by a third a party.

• In their simplest form, they are bilateral contracts between a buyer and seller where the seller sells protection against certain events occurring in relation to a third party who usually is not part of the contract.

• More complex forms of credit derivatives include Mortgage and other Synthetic Collateralized Debt Obligations (CDO).

• Securitization is the practice of pooling various debts such as mortgages, auto loans, credit card obligations, etc., and selling this consolidated debt as bonds, pass through securities, or Collateralized Mortgage Obligations (CMO).

Credit Derivative • Unfunded: When credit protection is bought and sold between

bilateral counterparties. – Each party is responsible for making its contract obligations

without the recourse to other assets. – No pool of assets or collateral that you can verify to cover a

default. – The protection seller makes no payments until there is a default.

• Funded: When the credit derivative is entered into by a financial institution or a special purpose vehicle (SPV), and payments under the credit derivative are funded using securitization techniques.

– The credit derivative will be embedded into a bond that is typically issued by an SPV or a financial institution.

– SPV, Special Purpose Vehicle, is typically a limited company or limited partnership created to fulfill a narrow or temporary objective, such as maintain the pool of assets or collateral.

– Bondholders will (ultimately) be responsible for the payment of any cash or physical settlement amounts.

Unfunded Credit Derivatives • Credit Default Swap (CDS): Buyer makes premium payments to seller;

seller makes payoff if the loan defaults. • Total Return Swap: Covers both credit and market risks allowing seller

to get all of the benefits of owning an asset without having the asset on the books; buyer gets the protection.

• Constant Maturity Credit Default Swap (CMCDS): Similar to a CDS, but premium payments are floating and based on a CDS index.

• Credit default swaption: This is an option with the right but not the obligation to enter into the swap.

• First to Default Credit Default Swap • Portfolio Credit Default Swap • Secured Loan Credit Default Swap • Credit Default Swap on Asset Backed Securities • Recovery lock transaction • Credit Spread Option • CDS index products • Constant Maturity Credit Default Swap (CMCDS)

Funded Credit Derivatives • Credit-linked notes CLN: Credit-linked note is a generic name

related to any bond whose value is linked to the performance of a reference asset, or assets.

• Collateralized debt obligation CDO: Generic term for a bond issued against a mixed pool of assets.

– There also exists CDO-squared (CDO2) where the underlying assets are CDO tranches.

– In 2007, credit rating agencies failed to account for the risk of nationwide collapse of home values when rating the corresponding CDOs.

• Collateralized bond obligations CBO: Bond issued against a pool of bond assets or other securities; often referred to in a generic sense as a CDO.

• Collateralized loan obligations CLO: Bond issued against a pool of bank loans; also is referred to in a generic sense as a CDO.

Value of a Credit Derivative Contract • In its simplest form, a credit derivative takes its value from the

underlying events of a third party.

Original Investor (Lender)

Borrower (counterparty)

Principal

Principal

InterestInsurance or Bank

Premium

Contingency payment

Credit Contract

Credit

Derivative

Contract

Questions on Credit Derivatives • What is value of the contract, or what premium should be

charged?

• What is the probability that the counterparty will default?

• What is the amount of capital the underwriter of the credit derivative should be holding to face a potential event that would trigger the contingency payment?

• How do we define an event? – When bankruptcy is filed? – When payments are missed? – When a credit agency downgrades the bond?

• How many contracts should an underwriter write?

• The potential exposure of contract c at time t it is defined as:

where T is the time to expiration of the contract and PV is the present value at time t.

• The actual exposure of contract c at time t is defined as:

where V(c,t) is the value of the contract at time t.

Credit Exposure • Credit exposure: The cost of replacing or hedging the contract at

the time of default. – It is the maximum value that will be lost if the counterparty

defaults.

( ) ( )[ ]{ }{ }tcVcVPVtcPE tTt ,,max,0max),( −= ≤< ττ

( )( , ) max 0, ,AE c t V c t=   

• Total exposure is the summation of actual and potential exposure.

Value of a Defaulted Security • Let’s look at an example of a zero coupon bond with one year to

maturity.

( )1 1 1 d d

PV P V P RV r

= − +  +

V

RV

1-Pd

Pd

PV

• With a certain probability (Pd), the bond defaults and recovers only a certain amount (RV) instead of the original payment (V).

• The problem with this approach is our assessment of this probability or risk.

– One way is to find a “risk neutral” (or arbitrage fee) estimate.

Example • Suppose that the par value of the previous bond is $1,000 and it is

currently trading at $930. – The current risk free rate is of 5% – This corresponds to a rate of return on the investment of

7.52%, which is significantly over the risk free rate. – We can considered the “excess” of return as the probability of

default. • Let’s assume that if the bonds defaults, the recovery value is zero.

– We therefore have:

• Solving we get Pd = 0.0235 • In some regards, this is similar to the Implied Volatility that we

have used in the past. • This approach also has some problems which we might explore in

a future lecture.

( ) ( )1 11 930 1 1000 0 1 1 0.05d d d d

PV p V p RV p p r

= − + = = − +      + +

Questions on Credit Derivatives • What is value of the contract, or what premium should be charged?

• What is the probability that the counterparty will default?

• What is the amount of capital the underwriter of the credit derivative should be holding to face a potential event that would trigger the contingency payment?

• How do we define an event? – When bankruptcy is filed? – When payments are missed? – When a credit agency downgrades the bond? – How does the definition of an event translate to contract value?

• How many contracts should an underwriter write?

Collateralized Debt Obligations • Collateralized debt obligations or CDOs are a form of credit

derivative offering exposure to a large number of companies in a single instrument.

– CDOs are funded credit derivatives. – This exposure is sold in slices of varying risk or

subordination – each slice is known as a tranche.

• In a cash flow CDO, the underlying credit risks are bonds or loans held by the issuer.

– Alternatively in a synthetic CDO, the exposure to each underlying company is a credit default swap.

– A synthetic CDO is also referred to as CSO.

• Other more complicated CDOs have been developed where each underlying credit risk is itself a CDO tranche.

– These CDOs are commonly known as CDOs-squared.

Implementation of Credit Link and CDOs

Cash flows before securitizationBorrowers

Originator

SPV

Investors

Cash flows after securitization

Sale price of securities Interest and principle

Profit extraction

Administrator Sale priceAsset sale

New home owners

Bank of America

Create a Special Purpose Vehicle “limited company” we can call “Mortgage XXX” that is an investment opportunity

Sell to investors as slices (tranches) with different levels of risk

Collateralized Debt Obligations • CDOs are constructed from a portfolio of fixed-income assets.

• These assets are divided into different tranches: – Senior tranches (rated AAA) – Mezzanine tranches (AA to BB) – Equity tranches (unrated).

• Losses or defaults are applied in reverse order of seniority. – So junior tranches offer higher coupons (interest rates) to

compensate for the added default risk.

• CDOs serve as an important funding vehicle for fixed-income assets.

Example • Suppose that the a local mortgage company (the originator)

issues 1000 mortgage balloon loans for a total of $1B at an average annual interest rate of 10%.

– All the mortgages expire in exactly ten years and cannot be repaid before maturity.

• This represents approximately $100M of annual interests for all the loans for the next 10 years.

• The mortgage company (the originator) sells these mortgages to another financial institution who bundles them as an investment package for investors.

– The original mortgages are offered as a collateral for the investment instrument.

• So, to make the investment instrument appealing to a large number of investors, the underlying collateral is segmented into “tranches” of different risks: Senior, Mezzanine and Equity.

• Each of the tranches is associated with a certain level of return.

Tranches • In general senior tranches are considered the least probable to default,

while the equity tranches the most likely to default. • From the previous example:

– Suppose that the top quality mortgages worth $200M are part of the senior tranche.

– The next $300M are part of the mezzanine tranche. – The remaining $500M are part of the equity tranche.

• Also assume that the interest rate offered to the senior tranche is 4%, 6% to the mezzanine tranche and 14.8% to the equity tranche.

• If no defaults occur, the expected interest streams are: $8M, $18M and $74M.

• Further assume that if any of the mortgages default, the income (or the lack thereof) will be taken from the equity first, then from the mezzanine, and last from the senior tranche.

• For example, suppose that only a total mortgage interest stream of $30M is obtained.

• Then the equity tranche would receive an annual interest stream of only $4M making the annual return of less than 1% per year.

Tranched Credit Risk • Achieving superior returns by introducing leverage in a credit

derivatives structure.

• The protection seller who commits to indemnify the protection buyer against the first 20% loss as a result of credit events, effectively has a leveraged position.

Credit Linked Notes • Credit-linked notes are frequently issued by special

corporations or trusts that hold a form of collateral securities financed through the issuer of the notes.

• The investor receives a coupon for taking the risk, as long as there has been no credit event of the reference entity.

• The corporation issuing the credit notes enters into a credit swap with a third party in which it sells default protection in return for a premium that subsidizes the coupon to compensate the investor for taking the risk.

Risk Management in the Insurance Industry

• Important terms: – Primary Layer: First level of insurance coverage above any

deductible. – Excess Layer: Level of Insurance coverage above the

primary layer. – Excess Coverage: Insurance that covers losses above an

attachment point below which there is usually another insurance policy or self insured retention.

– Umbrella Policy: In between primary and excess layer in an insurance program.

• A buffer layer is used when the umbrella policy requires underlying coverage limits that are higher than those provided by the primary layer.

Risk Assessment • Some of the main questions related to risk and CDOs include:

– What is the risk of the investment? – What is the appropriate return according to the level of risk? – What is the probability of different levels of default?

• Credit risk: Risk of loss that will be incurred in case of a default of a counterparty.

– The counterparty defaults when it fails to meet its contractual payments.

• To assess potential credit losses we need to: – Assess credit exposure of the counterparty at the time of the

default. – The probability of default. – The amount that can be recovered after the default.

Assignments • Watch the following videos (note the first video starts with a

black screen): – http://www.youtube.com/watch?v=oosYQHq2hwE&feature=related – http://www.youtube.com/watch?v=eYBlfxGIk28&feature=related – http://www.youtube.com/watch?v=q0oSKmC3Mfc&feature=related – http://www.youtube.com/watch?v=XjoJ9UF2hqg

• If you haven’t done so already, read The J.P. Morgan Guide to Credit Derivatives